The paradox of the tight spiral pass in American football: Insights from an analytic approximate solution

Price, Richard H.

doi:10.1119/10.0001632

In a recent paper, Price, Moss, and Gay have given a simple explanation of a paradox in the flight of a football, why the long axis of the football, contrary to intuition but in good agreement with experience, turns so that it is tangent to the path of the football. Here, we add to the analysis the assumption of only first-order differences between the direction of the velocity, the orientation of the long axis, and the direction of the total angular momentum. The result is a closed-form solution that is particularly useful in revealing the way in which nutation and precession are mixed.

INTRODUCTION

This short paper provides a supplementary view of the analysis described in a recent paper by Price, Moss, and Gay¹ (hereafter PMG). The target of that paper, and of this paper, is the rotational kinematics and dynamics of a spiraling football during its trajectory. The particular focus is the clarification of how the symmetry axis of the axisymmetric American football remains closely tangent to the path of the football center of mass (CM).

In comparison with PMG, in this paper, there is a trade-off. At the cost of an additional (but very well justified) approximation, we achieve a closed form (“analytic”), rather than numerical, solution. As is usually the case, the closed form solution gives insights not available from a numerical solution. For the spiral pass, the insights are particularly useful in distinguishing precession and nutation.

We use the same notation as in PGM, notation that is similar to that of Soodak² and that is briefly reviewed as follows:

L = Football angular momentum= $L \hat{l}$ ⁠, where L = magnitude of L,
$\hat{s}$ = Unit vector aligned with the symmetry axis of the football,
$\hat{v}$ = Unit vector in the direction of the velocity of the football CM,
I_t = Moment of inertia about an axis through the CM and perpendicular to $\hat{s}$ ⁠,
$τ^{*}$ = Magnitude of the aerodynamic torque in a simple model introduced by Soodak, experimentally verified by Rae and Streit,³ and presented below.

ANALYSIS

We start with the two rotational equations that are the foundation of the analysis in PMG,

\dot{\hat{s}} \equiv \frac{1}{I_{t}} L \times \hat{s} and \dot{L} = τ^{*} \hat{v} \times \hat{s},

(1)

where the overdot indicates a time derivative. The first of these is an equation of rotational kinematics that is somewhat arcane but is relatively easy to derive. The derivation is given by Soodak² and is repeated in the appendix of the PMG paper. The second is a model of aerodynamic torque, the “pitch torque,” which again is given by Soodak and discussed in PMG.

We now make the approximation that

\hat{l}, \hat{s},

and

\hat{v}

differ only by small amounts, and we expand Eqs. to first order in those small amounts. Specifically, we define⁴

δ \equiv \hat{s} - \hat{v} and ε \equiv \hat{l} - \hat{v},

(2)

and we expand to first order in

δ

and

ε

⁠. (The numerical results in PMG are one justification for the appropriateness of this approximation for a well thrown spiral pass.)

Some simplification follows immediately from taking the dot product of L with the second of Eqs. and using

L \cdot \dot{L} = 1 / 2 d (L^{2}) / d t

⁠,

\frac{1}{2} \frac{d (L^{2})}{d t} = L τ^{*} (\hat{v} \times \hat{s}) \cdot \hat{l} = L τ^{*} (\hat{v} \times [δ + \hat{v}]) \cdot [ε + \hat{v}] = L τ^{*} (\hat{v} \times δ) \cdot ε .

(3)

It follows that the time derivative of L is second order in our perturbation quantities, and henceforth, we can consider L to be constant in time. We can thus rewrite Eqs. as

\dot{\hat{s}} = ω_{wob} \hat{l} \times \hat{s} and \dot{\hat{l}} = ω_{gyr} \hat{v} \times \hat{s},

(4)

with

ω_{wob} \equiv \frac{L}{I_{t}} and ω_{gyr} \equiv \frac{τ^{*}}{L} .

(5)

Our next step is to eliminate

\hat{s}

and

\hat{l}

in favor of

\hat{v}

and the perturbations. This gives

\dot{δ} = ω_{wob} (\hat{v} \times δ + ε \times \hat{v}) - \dot{\hat{v}},

(6)

\dot{ε} = ω_{gyr} (\hat{v} \times δ) - \dot{\hat{v}} .

(7)

An important and intuitively appealing conclusion follows:

\dot{\hat{v}}

must be first order in the perturbation order. By comparing terms on the right in Eqs. and , we see that the rotation rate of the football divided by either

ω_{wob}

or

ω_{gyr}

must be of first order; for

\hat{l}, \hat{s}, \hat{v}

to remain well aligned, the change in the direction of the football's motion must be slow.

Equations and are first-order differential equations for the time development of

δ

and

ε

once we specify the (slow!) time dependence of

\hat{v}

⁠. To clarify the nature of this system of equations, we start by writing out the components of the equations in the same {x, y, z} coordinate system used in the PMG paper. (The football trajectory is confined to the x, z plane, with x directed vertically upward.) The component equations are

{\dot{δ}}_{x} = ω_{wob} (- {\hat{v}}_{z} δ_{y} + {\hat{v}}_{z} ϵ_{y}) - {\dot{\hat{v}}}_{x},

(8)

{\dot{δ}}_{y} = ω_{wob} ({\hat{v}}_{z} δ_{x} - {\hat{v}}_{x} δ_{z} - {\hat{v}}_{z} ϵ_{x} + {\hat{v}}_{x} ϵ_{z}),

(9)

{\dot{δ}}_{z} = ω_{wob} ({\hat{v}}_{x} δ_{y} - {\hat{v}}_{x} ϵ_{y}) - {\dot{\hat{v}}}_{z},

(10)

{\dot{ϵ}}_{x} = ω_{gyr} {\hat{v}}_{z} δ_{y} - {\dot{\hat{v}}}_{x},

(11)

{\dot{ϵ}}_{y} = ω_{gyr} ({\hat{v}}_{z} δ_{x} - {\hat{v}}_{x} δ_{z}),

(12)

{\dot{ϵ}}_{z} = ω_{gyr} {\hat{v}}_{x} δ_{y} - {\dot{\hat{v}}}_{z} .

(13)

We are interested in the projections of the symmetry axis and angular momentum in the plane perpendicular to the velocity, so we use the unit vectors,

\hat{Y} = \hat{y} and \hat{X} = {\hat{v}}_{z} \hat{x} - {\hat{v}}_{x} \hat{z},

(14)

introduced in PMG, Sec. III, and we define the “deviation vectors” S, E, in that plane as the vectors with the components,

S_{X} = δ \cdot \hat{X}, S_{Y} = δ \cdot \hat{Y}, E_{X} = ε \cdot \hat{X}, E_{Y} = ε \cdot \hat{Y} .

(15)

It should be noted that this S is the same as in PMG and that E is the analogous vector for L. From Eqs. , we arrive at the equations for the time derivatives of these quantities,

{\dot{S}}_{X} = ω_{wob} (- S_{Y} + E_{Y}) + {\hat{v}}_{x} {\dot{\hat{v}}}_{z} - {\hat{v}}_{z} {\dot{\hat{v}}}_{x} + δ_{x} {\dot{\hat{v}}}_{z} - δ_{z} {\dot{\hat{v}}}_{x},

(16)

{\dot{S}}_{Y} = ω_{wob} (S_{X} - E_{X}),

(17)

{\dot{E}}_{X} = ω_{gyr} S_{Y} + {\hat{v}}_{x} {\dot{\hat{v}}}_{z} - {\hat{v}}_{z} {\dot{\hat{v}}}_{x} + ϵ_{x} {\dot{\hat{v}}}_{z} - ϵ_{z} {\dot{\hat{v}}}_{x},

(18)

{\dot{E}}_{Y} = ω_{gyr} S_{X} .

(19)

The products of

δ

and

\dot{\hat{v}}

at the end of Eq. and products of

ε

and

\dot{\hat{v}}

at the end of Eq. are second order in the perturbations and will henceforth be ignored. The terms,

Ω (t) \equiv {\hat{v}}_{x} {\dot{\hat{v}}}_{z} - {\hat{v}}_{z} {\dot{\hat{v}}}_{x},

(20)

in Eqs. and are the y component of

\dot{\hat{v}} \times \hat{v}

and represent the rate of rotation of the football velocity in the xz plane.

This system of equations can be put in matrix form

[\begin{matrix} \dot{S_{X}} \\ \dot{S_{Y}} \\ \dot{E_{X}} \\ \dot{E_{Y}} \end{matrix}] = [\begin{matrix} 0 & - ω_{wob} & 0 & ω_{wob} \\ ω_{wob} & 0 & - ω_{wob} & 0 \\ 0 & ω_{gyr} & 0 & 0 \\ ω_{gyr} & 0 & 0 & 0 \end{matrix}] [\begin{matrix} S_{X} \\ S_{Y} \\ E_{X} \\ E_{Y} \end{matrix}] + [\begin{matrix} Ω (t) \\ 0 \\ Ω (t) \\ 0 \end{matrix}] .

(21)

These equations represent the driving of

δ

and

ε

by the slow rotation of the velocity. The matrix can be very simply inverted, and the general solution is expressed in terms of the driving term

Ω (t)

and the homogeneous solutions of Eq. .

For the greatest clarity of the solution, we now make our second assumption: we assume a constant rate of rotation and write this as

{\hat{v}}_{z} = cos Ω_{0} t and {\hat{v}}_{x} = - sin Ω_{0} t,

(22)

as in PMG Sec. VI. (We have inserted the zero subscript to help distinguish the constant rotation rate from the more general rotation rate Ω.) For the long bomb paradigmatic pass of PMG, it is pointed out in Sec. VI that the variation of Ω is significant but less than around 20%.

We can now remove the source on the right in Eq. by replacing the S and E components with

S_{X} = {\tilde{S}}_{X}; S_{Y} = {\tilde{S}}_{Y} + Ω_{0} / ω_{gyr}; E_{X} = {\tilde{E}}_{X}; E_{Y} = {\tilde{E}}_{Y} + Ω_{0} / ω_{gyr} - Ω_{0} / ω_{wob} .

(23)

Then, the equations are those of Eq. for the quantities with tildes and with no source terms. The eigenvalues and eigenvectors of the matrix in Eq. represent the four independent homogeneous solutions of those equations, and the four solutions in Eq. are what we need to keep track of all the kinematic quantities

v

⁠, L, and

\hat{s}

⁠, four functions because there are four degrees of freedom in the kinematics: The velocity is a specified vector, and the magnitude of L is fixed by its initial value, so only the four degrees of freedom of

\hat{s}

and

\hat{l}

⁠, in the plane perpendicular to

\hat{v}

⁠, need to be computed. If we are given the initial values of

\hat{s}

and

\hat{l}

⁠, their values at any subsequent time will be the superposition of the four solutions in Eq. .

With Mathematica, the solutions of Eq. (21) with the velocity in Eq. (22) are easily found to be as follows:

Solution F1:

{\tilde{S}}_{X}^{H} = - 2 ω_{wob}^{2} ω_{fst} sin (ω_{fst} t), {\tilde{S}}_{Y}^{H} = ω_{wob} (ω_{wob}^{2} + χ) cos (ω_{fst} t),

(24)

{\tilde{E}}_{X}^{H} = ω_{fst} (χ - ω_{wob}^{2}) sin (ω_{fst} t), {\tilde{E}}_{Y}^{H} = 2 ω_{gyr} ω_{wob}^{2} cos (ω_{fst} t) .

(25)

Solution F2:

{\tilde{S}}_{X}^{H} = - 2 ω_{wob}^{2} ω_{fst} cos (ω_{fst} t), {\tilde{S}}_{Y}^{H} = - ω_{wob} (ω_{wob}^{2} + χ) sin (ω_{fst} t),

(26)

{\tilde{E}}_{X}^{H} = ω_{fst} (χ - ω_{wob}^{2}) cos (ω_{fst} t), {\tilde{E}}_{Y}^{H} = - 2 ω_{gyr} ω_{wob}^{2} sin (ω_{fst} t) .

(27)

Solution S1:

{\tilde{S}}_{X}^{H} = - 2 ω_{wob}^{2} ω_{slw} sin (ω_{slw} t), {\tilde{S}}_{Y}^{H} = ω_{wob} (ω_{wob}^{2} - χ) cos (ω_{slw} t),

(28)

{\tilde{E}}_{X}^{H} = - ω_{slw} (χ + ω_{wob}^{2}) sin (ω_{slw} t), {\tilde{E}}_{Y}^{H} = 2 ω_{gyr} ω_{wob}^{2} cos (ω_{slw} t) .

(29)

Solution S2:

{\tilde{S}}_{X}^{H} = - 2 ω_{wob}^{2} ω_{slw} cos (ω_{slw} t), {\tilde{S}}_{Y}^{H} = - ω_{wob} (ω_{wob}^{2} - χ) sin (ω_{slw} t),

(30)

{\tilde{E}}_{X}^{H} = - ω_{slw} (χ + ω_{wob}^{2}) cos (ω_{slw} t), {\tilde{E}}_{Y}^{H} = - 2 ω_{gyr} ω_{wob}^{2} sin (ω_{slw} t) .

(31)

(Note that the homogeneous solutions

{\tilde{S}}^{H}, {\tilde{E}}^{H}

are not dimensionally the same as

\tilde{S}, \tilde{E}

⁠. In the application to a problem, the latter are constructed from the former with coefficients that have dimension (time)³.) In the above equations,

χ \equiv \sqrt{- 4 ω_{gyr} ω_{wob}^{3} + ω_{wob}^{4}},

(32)

and the fast and slow eigenfrequencies are

(33)

ω_{fst} \equiv \sqrt{- ω_{gyr} ω_{wob} + \frac{ω_{wob}^{2}}{2} + \frac{1}{2} χ},

(33a)

ω_{slw} \equiv \sqrt{- ω_{gyr} ω_{wob} + \frac{ω_{wob}^{2}}{2} - \frac{1}{2} χ} .

(33b)

Figure 1 shows the result of the closed-form solution in Eqs. (24)–(33) for a “long bomb” spiral pass and compares it with the numerical solutions of Eqs. (1) for the long bomb spiral pass of PMG (Table I), with its true parabolic trajectory, solved as a coupled set of differential equations. In view of the fact that the initial misalignment is not particularly small (10°) and that the rotational rate of the true long bomb is not constant, the agreement is remarkable.

The system in becomes even simpler when we invoke the condition

ω_{gyr} ≪ ω_{wob}

that describes a well thrown football pass. In this case, to the lowest order in

ω_{gyr} / ω_{wob}

⁠, we have

ω_{fst} \to ω_{wob}, ω_{slw} \to ω_{gyr} .

(34)

Approximate solution F1:

{\tilde{S}}_{X}^{H} = - 2 ω_{wob}^{3} sin (ω_{wob} t), {\tilde{S}}_{Y}^{H} = 2 ω_{wob}^{3} cos (ω_{wob} t),

(35)

{\tilde{E}}_{X}^{H} = 2 ω_{gyr} ω_{wob}^{2} sin (ω_{wob} t), {\tilde{E}}_{Y}^{H} = 2 ω_{gyr} ω_{wob}^{2} cos (ω_{wob} t) .

(36)

Approximate solution F2:

{\tilde{S}}_{X}^{H} = - 2 ω_{wob}^{3} cos (ω_{wob} t), {\tilde{S}}_{Y}^{H} = - 2 ω_{wob}^{3} sin (ω_{wob} t),

(37)

{\tilde{E}}_{X}^{H} = 2 ω_{gyr} ω_{wob}^{2} cos (ω_{wob} t), {\tilde{E}}_{Y}^{H} = - 2 ω_{gyr} ω_{wob}^{2} sin (ω_{wob} t) .

(38)

Approximate solution S1:

{\tilde{S}}_{X}^{H} = - 2 ω_{gyr} ω_{wob}^{2} sin (ω_{gyr} t), {\tilde{S}}_{Y}^{H} = 2 ω_{gyr} ω_{wob}^{2} cos (ω_{gyr} t),

(39)

{\tilde{E}}_{X}^{H} = - 2 ω_{gyr} ω_{wob}^{2} sin (ω_{gyr} t), {\tilde{E}}_{Y}^{H} = 2 ω_{gyr} ω_{wob}^{2} cos (ω_{gyr} t) .

(40)

Approximate solution S2:

{\tilde{S}}_{X}^{H} = - 2 ω_{gyr} ω_{wob}^{2} cos (ω_{gyr} t), {\tilde{S}}_{Y}^{H} = - 2 ω_{gyr} ω_{wob}^{2} sin (ω_{gyr} t),

(41)

{\tilde{E}}_{X}^{H} = - 2 ω_{gyr} ω_{wob}^{2} cos (ω_{gyr} t), {\tilde{E}}_{Y}^{H} = - 2 ω_{gyr} ω_{wob}^{2} sin (ω_{gyr} t) .

(42)

CONCLUSIONS

From these results, we see that the fast precession solutions F1 and F2 describe a motion in which $ε$ is smaller than $δ$ by the ratio $ω_{gyr} / ω_{wob}$ ⁠, which means that the alignment of $\hat{l}$ and $\hat{v}$ is much better than that of $\hat{s}$ and $\hat{v}$ ⁠. This is just what we expect for quasi torque-free precession.

When the motion is that of S1 and S2, we see that S and E are the same, meaning that $δ$ and $ε$ are the same. Thus, for $ω_{gyr} ≪ ω_{wob}$ ⁠, the symmetry axis remains well aligned with the angular momentum as they both rotate at $ω_{gyr}$ due to the aerodynamic torque.

One of the advantages of a closed form solution is the insight into the interaction of the “torqued precession” frequency $ω_{gyr}$ and the “nutation” frequency $ω_{wob}$ ⁠. The fast and slow frequencies of Eqs. (33) show that for $ω_{gyr} ≪ ω_{wob}$ ⁠, as in the case of our “long pass,” the “fast” oscillation is slightly less than $ω_{wob}$ and the “slow” oscillation is slightly greater than $ω_{gyr}$ ⁠.

ACKNOWLEDGMENTS

The author thanks the coauthors of PMG, William Moss and Timothy Gay, for their patience and tolerance.

REFERENCES

1.

R. H.

Price

,

W. C.

Moss

, and

T. J.

Gay

, “

The paradox of the tight spiral pass in American football: A simple resolution

,”

Am. J. Phys.

(in press); see the references in that paper for background literature

https://doi.org/10.1119/10.0001388

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H.

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The current paper can be considered to be an extension of the “Simplified Analytical Model” in Sec. VI of PMG, with the assumption dropped that $\hat{s}$ and $\hat{l}$ differ negligibly, so that there is no difference between $δ$ and $ε$ ⁠. This reduction of the number of degrees of freedom preserves the resolution of the spiral pass paradox but eliminates the high frequency nutation.

2020

American Association of Physics Teachers

The paradox of the tight spiral pass in American football: Insights from an analytic approximate solution

INTRODUCTION

ANALYSIS

CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

Contents

Figures & Tables

Supplementary Material

References

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The paradox of the tight spiral pass in American football: Insights from an analytic approximate solution

INTRODUCTION

ANALYSIS

CONCLUSIONS

ACKNOWLEDGMENTS

REFERENCES

Contents

Figures & Tables

Supplementary Material

References

Related

Citing articles via

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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